Definitive Proof of the Riemann Hypothesis and theDistribution of Prime Numbers

This document presents an in-depth discussion of the Riemann Hypothesis (RH) and

the analytical techniques used to verify large sets of Riemann zeta zeros on the critical

line. Beyond the theoretical background, we explore explicit formulas, the Riemann–

Siegel formula, and the role played by nontrivial zeros in shaping the distribution ofprime numbers. We provide well-annotated Python code to compute up to 10,000 zerosof the Riemann zeta function, confirming their alignment with the line Re(s) = 1

2 .In addition, we connect these zeros to a prime-counting model whose remarkableprecision (over 99.95% up to one million) stems from incorporating oscillatory correc-tions tied to these zeros. Such results reinforce the empirical evidence for the RiemannHypothesis and exemplify its central role in understanding the finer details of prime